Alternative q-Analog

Question

Wikipedia gives the following definition of a q-analog: "In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter $q$ that returns the original theorem, identity or expression in the limit as $q \rightarrow 1$. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results."

There seem to exists, for basic mathematical objects like; the numbers, factorial, binomial coefficient,... a well-known q-analog established as convention, however for other lesser known objects like specific types of numbers, or derivatives, there seem to be a few different types of q-analogs all with their own unique properties. These q-analogs are all still based off of the 'basic building block' of all q-analogs, which is the q-number

$${[n]}_q = \sum_{k=0}^{n-1}q^k$$ Which has the arithemtic properties $${[n m]}_q = {[n]}_q{[m]}_{q^n}$$ and $${[n+m]}_q = {[n]}_q+q^n{[m]}_q$$

My question is just if there are other known q-analogs of, for example the number. Perhaps even known q-analogs that return the original theorem when $q \rightarrow x$ for some $x \neq 1$? Of course anyone could arbitrarily create a q-analog like $n^q$ so when $q \rightarrow 1$ returns $n$ but that doesn't seem that useful. Any contribution is appreciated

---

Edit: Just to add this here in case anyone is interested. I found something called a d-analog from this paper here. Regardless of the notation, this is relevant to the post, and since it doesn't seem very popular I thought I could share it here and maybe someone finds it interesting.

Bounty

A +100 bounty is set for anyone who can create/showcase a well-defined and interesting alternative analog for some basic 'building-block' mathematical object like a number $n \in \mathbb{Z}$. You can go into as much detail, perhaps even formulating $n!$, analog derivatives, $n \in \mathbb{Q}$, showcasing how it is useful, etc. Bounty goes to the best one! Good luck!

Answer 1

The question is about alternative $q$-analogs.

My question is just if there are other known q-analogs of, for example the number.

As stated in the question itself

Of course anyone could arbitrarily create a q-analog like $n^q$ so when $q \rightarrow 1$ returns $n$ but that doesn't seem that useful.

One important arithmetic property of the integers is the divisibility relation. The well-known $q$-analog $\,[n]_q\,$ has this divisibility property. It is just one example of a divisibility sequence. From Wikipedia:

In mathematics, a divisibility sequence is an integer sequence $(a_{n})$ indexed by positive integers $n$ such that $$ {\text{if }}m\mid n{\text{ then }}a_{m}\mid a_{n} $$ for all $m, n$. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.

After this is a list of examples of integer divisibility sequences. The last two examples are

• More generally, any Lucas sequence of the first kind $U_n(P,Q)$ is a divisibility sequence. Moreover, it is a strong divisibility sequence when $\gcd(P,Q) = 1$.
• Elliptic divisibility sequences are another class of such sequences.

The $q$-analog $\,[n]_q = U_n(1,q)\,$ is a special case of Lucas sequences. Given a divisibility sequence, there is associated an analog of the factorials and binomial coefficients. For example, Fibonomial coefficients are based on the Fibonacci sequence which is a special case of a Lucas sequence.

This analog has applications as in the 2020 paper "Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers". There is an earlier 2009 paper "Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences" by Sagan and Savage.

Another different generalization of $\,[n]_q\,$ comes from the theory of elliptic hypergeometric functions. For example, one definition of elliptic numbers is from a 2020 paper "Elliptic and $q$-Analogs of the Fibonomial Numbers" by Bergeron, Ceballos and Kustner. First define the notations

$$ (a;p)_\infty := \prod_{k=0}^\infty(1-ap^k), \quad (a_1,\dots,a_m;p)_\infty := (a_1;p)_\infty\cdots(a_m;p)_\infty $$

and

$$ \theta(x;p) := (x,p/x;p)_\infty, \quad \theta((a_1,\dots,a_m;p)_\infty := \theta(a_1;p)_\infty \cdots\theta(a_m;p)_\infty. $$

Their definition of elliptic number is

$$ [n]_{a,b,q,p} := \frac{\theta(q^n,aq^n,bq,\frac{a}{b}q;p)} {\theta(q,aq,bq^n,\frac{a}{b}q^n;p)}. $$

Further combinatorial applications is from a 2017 paper "Elliptic rook and file numbers" by Schlosser and Yoo. They have the formulas

$$ \theta(x;p) = - x\theta(1/x;p), \qquad \theta(px;p) = \theta(x;p)/x. $$

and the addition formula

$$ \theta(xy,x/y,uv,u/v;p) - \theta(xv,x/v,uy,u/y;p) = \theta(yv,y/v,xu,x/u;p)u/y. $$

Their definition of elliptic number is slightly different:

$$ [z]_{a,b;q,p} = \frac{\theta(q^z,aq^z,bq^2,a/b;p)} {\theta(q,aq,bq^{z+1},aq^{q-1}/b,p)}. $$